Largest Area River B94C69

1. **State the problem:** A farmer wants to enclose a rectangular plot next to a river using 3000 feet of fencing. Since the river forms one side, fencing is needed for only three sides. 2. **Define variables:** Let $x$ be the length of the side parallel to the river (no fencing needed on the river side). Let $y$ be the length of each of the two other sides perpendicular to the river. 3. **Write the fencing constraint:** Since the river side needs no fence, the total fencing used is for two sides of length $y$ and one side of length $x$: $$2y + x = 3000$$ 4. **Express $x$ in terms of $y$:** $$x = 3000 - 2y$$ 5. **Write the area function:** The area $A$ of the rectangle is: $$A = x \times y$$ Substitute $x$: $$A = (3000 - 2y) y = 3000y - 2y^2$$ 6. **Maximize the area:** To find the maximum area, take the derivative of $A$ with respect to $y$ and set it to zero: $$\frac{dA}{dy} = 3000 - 4y = 0$$ 7. **Solve for $y$:** $$3000 - 4y = 0 \implies 4y = 3000 \implies y = \frac{3000}{4} = 750$$ 8. **Find $x$ using $y=750$:** $$x = 3000 - 2(750) = 3000 - 1500 = 1500$$ 9. **Calculate the maximum area:** $$A = x \times y = 1500 \times 750 = 1125000$$ 10. **Conclusion:** The largest area that can be enclosed is $$1125000$$ square feet when the side parallel to the river is $$1500$$ feet and the other two sides are each $$750$$ feet.